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A First Course in Number Theory

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Author(s): Alexandru Buium
Series: lecture notes
Edition: version 2019-04-26
Year: 2019

Language: English
Commentary: Downloaded from https://math.unm.edu/~buium/unt.pdf

1. Introduction 2
2. The integers 4
3. Induction 6
4. Finite sets, finite sums, finite products 7
5. The rationals 8
6. Divisibility and Euclid division 10
7. Polynomial time algorithms 12
8. Primes 13
9. Greatest common divisor 14
10. Unique factorization 15
11. Applications of unique factorization 17
12. Congruences: generalities 19
13. Complete residue systems 20
14. Residue classes 21
15. Inverses mod m 22
16. Groups 23
17. Linear congruences 25
18. Systems of linear congruences 25
19. Fermat’s little theorem 26
20. Euler’s theorem 27
21. Polynomial congruences 28
22. Langrange’s theorem 30
23. Order 31
24. Primitive roots 32
25. Discrete logarithm 33
26. Legendre symbol 35
27. Gaussian integers 37
28. Fundamental Theorem of Arithmetic for Gaussian integers 38
29. Factoring prime integers in the Gaussian integers 39
30. Real and complex numbers 40
31. Algebraic integers 40
32. Non-unique factorization in Kummer integers 42
33. Proof of Quadratic Reciprocity 43
34. Appendix: Cryptography 45